The gravity equation most people are looking for is Newton’s Law of Universal Gravitation: F = G × (m₁ × m₂) / r². It calculates the gravitational force between any two objects in the universe using just their masses and the distance between them. This single formula explains everything from why apples fall to how planets orbit stars, and it remains the standard equation for gravity in most practical applications more than 300 years after Newton published it.
Newton’s Law of Universal Gravitation
The full equation is:
F = G × (m₁ × m₂) / r²
Each variable represents something straightforward:
- F is the gravitational force between the two objects, measured in newtons.
- G is the universal gravitational constant, a fixed number that makes the units work out. Its value is 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², as determined by the 2022 CODATA measurement published by NIST.
- m₁ and m₂ are the masses of the two objects, in kilograms.
- r is the distance between the centers of the two objects, in meters.
Two key relationships are built into this equation. First, gravity scales directly with mass. Double the mass of either object and the force doubles. Second, gravity weakens with the square of the distance. Move two objects twice as far apart and the gravitational pull drops to one quarter. This “inverse square” behavior is why you can feel Earth’s gravity powerfully at the surface but astronauts in orbit, only a few hundred kilometers higher, experience near-weightlessness.
The gravitational constant G is notably tiny, which is why gravity only becomes noticeable when at least one object is astronomically massive. Two bowling balls sitting a meter apart pull on each other with a force far too small to detect without laboratory instruments. But replace one bowling ball with Earth (5.97 × 10²⁴ kg), and the force becomes what you experience as weight.
The Simplified Equation for Objects Near Earth
When you’re working with objects near Earth’s surface, a simpler version of the gravity equation handles most situations. Since one of the two masses is always Earth and the distance is always roughly Earth’s radius, those values get bundled into a single number: g, the acceleration due to gravity. Its value is 9.81 m/s².
This gives you two useful simplified equations depending on what you’re calculating:
Weight: F = m × g. Your mass in kilograms times 9.81 gives your weight in newtons. An 80 kg person weighs about 785 newtons at sea level.
Free fall: Any object dropped near Earth’s surface accelerates downward at 9.81 m/s² regardless of its mass, as long as air resistance is negligible. A feather and a hammer dropped in a vacuum hit the ground at the same time. The equations for a falling object are v = g × t (speed equals gravity times time) and d = ½ × g × t² (distance fallen equals half of gravity times time squared). After one second of free fall, an object is moving at 9.81 m/s and has fallen about 4.9 meters.
Gravitational Potential Energy
Gravity doesn’t just pull things down. It also stores energy in objects based on their height. The equation for gravitational potential energy near Earth’s surface is:
PE = m × g × h
Here, m is mass, g is 9.81 m/s², and h is the height above whatever reference point you choose (usually the ground). A 70 kg person standing on a 10-meter diving platform has about 6,867 joules of gravitational potential energy relative to the water below. The moment they jump, that stored energy converts into kinetic energy as they accelerate downward. The path doesn’t matter: whether you slide down a ramp or fall straight, the same height change produces the same energy change.
How Surface Gravity Differs Across Worlds
The gravity equation explains why you’d weigh different amounts on different planets or moons. Surface gravity depends on two things: how massive the body is and how large its radius is. A planet can be less massive than Earth but still have similar surface gravity if it’s also much smaller (putting you closer to its center of mass).
The Moon is a clear example. It has about 1/81st of Earth’s mass, but it’s also much smaller in radius. The net result is a surface gravity roughly 1/6th of Earth’s. A person weighing 180 pounds on Earth would weigh about 30 pounds on the Moon. Mars sits in between, with a surface gravity about 38% of Earth’s. Jupiter, despite being a gas giant with no solid surface, has a gravitational pull about 2.4 times stronger than Earth’s at its cloud tops.
This same principle determines the boundary between Earth’s and the Moon’s gravitational influence. On a trip from Earth to the Moon, there’s a point where the Moon’s weaker but closer gravitational pull overtakes Earth’s stronger but more distant pull, and a spacecraft begins drifting toward the lunar surface instead.
Where Newton’s Equation Breaks Down
Newton’s gravity equation works extraordinarily well for nearly every situation you’ll encounter, from engineering to space missions. It even led to the discovery of Neptune in 1846, when astronomers noticed Uranus wasn’t following its predicted orbit and calculated where an unseen planet must be tugging on it. But the equation has limits.
The most famous failure involves Mercury. Mercury’s orbit shifts slightly over time, with its closest point to the Sun rotating around in a slow loop called precession. Newton’s equation, accounting for the gravitational influence of all known planets, predicted most of this shift but left a 43 arcsecond per century discrepancy that it simply couldn’t explain. That tiny error stood unresolved for decades until Einstein’s general theory of relativity, published in 1915, predicted the extra precession exactly.
Einstein replaced the concept of gravity as a force with gravity as the curvature of space and time caused by mass and energy. Near extremely massive or dense objects, or at speeds approaching the speed of light, Newton’s equation gives increasingly inaccurate results. GPS satellites, for instance, require corrections based on general relativity to stay accurate. Without them, position errors would accumulate by roughly 10 kilometers per day.
Gravity at the Extreme: Black Holes
The gravity equation also points toward one of the strangest objects in the universe. If you compress enough mass into a small enough space, gravity becomes so strong that nothing, not even light, can escape. The boundary beyond which escape is impossible is called the event horizon, and its size is given by the Schwarzschild radius:
R = 2GM / c²
Here, G is the same gravitational constant from Newton’s equation, M is the object’s mass, and c is the speed of light. The relationship is remarkably simple: the radius scales directly with mass. A black hole with 10 times the mass has an event horizon 10 times larger. For a black hole with the mass of our Sun, the Schwarzschild radius works out to about 3 kilometers. Earth compressed into a black hole would have an event horizon smaller than a marble, roughly 9 millimeters across.